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The thinking behind Citizen Maths

Once Citizen Maths got underway, people interested in the learning and teaching of maths started to ask us to explain our thinking. Here is Dave Pratt’s and Seb Schmoller’s overview.

Who the course is for

Citizen Maths is a free open online maths course for:

self-motivated individuals whose level of mathematical capability is at or above NVQ Level 1, but is not yet at NVQ Level 3, and who want to improve it [Note for international readers: Level 2 is the level that 16 year old school leavers are expected to achieve. About 60% do so.];

employers who want to provide staff (or trade unions their members) with a practical and flexible learning and development opportunity in maths;

colleges and other learning providers who want to give enrolled learners an additional or alternative route to improving their maths.

Learners will need to have use of (and know how to use) a desktop or laptop computer with a broadband internet connection.

Free and open

Citizen Maths — the first part of which was launched in September 2014, with the full course completed in June 2016 — is a course made available over the Internet without charge. If, as we hope, thousands of learners use the course, it will also warrant being referred to as a massive open online course, or MOOC. We’ve modelled Citizen Maths in part on the 2011 artificial intelligence (AI) MOOC designed and run by Google’s Director of Research Peter Norvig and Stanford University’s Sebastian Thrun. Our aim is to give learners the feeling that they are in a one-to-one tutorial. So Citizen Maths is made of very short instructional videos by our two “to-camera” tutors, who are very experienced maths teachers. However, Citizen Maths has features not in the AI MOOC. For example:

we provide purpose-made app-based activities to help learners build up their mathematical understanding, including a small amount of programming, using Scratch;

running in tandem with Google Coursebuilder, we intend, from 2015, to draw on the capabilities of the CogBooks adaptive learning system, which results in learners taking different paths through the course, depending on how they are doing.

Helping to tackle an otherwise insoluble problem

Data from the OECD’s 2013 “PIAAC” Skills Report [last accessed 8 June 2014], shows that about 1 in 3 of the UK’s adult population – say ten million people – have a current level of mathematical capability that would enable them to benefit from Citizen Maths. This represents a challenge that is very difficult to address through traditional methods of learning and teaching: many are disenfranchised by life circumstances from taking part in face-to-face courses; and the challenge would be also be very costly to solve conventionally. Of course, nothing like all people will have the necessary self-motivation, ICT access and ICT skills to use Citizen Maths. But the absolute number of people in the UK population for whom Citizen Maths should be suitable, is nevertheless large; and, if Citizen Maths is successful with learners, we will have made a contribution to solving the “intermediate level” maths challenge, at a low enough cost per learner for the course (and similar courses) to be offered more widely.

Mathematics considerations

The need for a new approach

Typically, our learners will have been through the conventional education system and they will have been failed by that system, coming out of it with qualifications that they now feel do not reflect what they might have achieved, given different opportunities, and what they now need, either for personal satisfaction or for more utilitarian purposes. Like many others, they probably do not see the point of maths, find it hard to engage with the subject, and simply ‘glaze-over’ when presented with numbers. As a result, perhaps they feel a sense of disempowerment, captured memorably by the astronomer, Carl Sagan (1996) in his final interview, who argued that, without scientific understanding, “we don’t run the government, the government runs us”. This view could equally be applied to mathematical understanding.

It is a key starting point for the design of Citizen Maths that these individuals will benefit from engaging with a different approach to the teaching and learning of mathematics from the conventional pedagogy that did not work for them in the past. Our design sets out to avoid portraying maths as something abstract. Instead it puts maths in the context of many adults’ everyday lives.

Engaging with contextualised problems

Citizen Maths turns learning mathematics on its head. Rather than present mathematics as a sequence of apparently disconnected routines and procedures, it engages people in familiar activity to reveal the ‘maths inside’ and it gives access to its power. It exploits the fact that there is probably no other area of education that has such an immediate relevance to the problems we all of us have to solve every day. These problems could range from comparing deals and prices on groceries and creating a household budget, to understanding a payslip, creating sales forecasts, keeping track of savings and pensions, controlling a production process, or making political judgements.

By the careful choice of problems set in meaningful contexts, learners who follow the Citizen Maths course will begin to recognise the power of key mathematical ideas. We claim that it is this sense of the power of mathematics that makes the discipline meaningful and it is exactly this feeling that has previously evaded learners, giving them the illusory perception that mathematics is abstract, meaningless and irrelevant to their lives.

Powerful ideas in action

“Learning results from what the student does and thinks and only from what the student does and thinks. The teacher can advance learning only by influencing what the student does to learn.”

In Citizen Maths we apply the above “axiom”, which is attributed to Nobel Prize winner Herbert Simon. We’ve designed the course so that learners engage in contextualised problems in such a way that the power of mathematics is revealed. This is what we mean by powerful ideas in action. For example, at Level 2, the concept of proportion underlies fraction, percentage, decimals, ratio, and probability, to name but a few maths ideas. Although it is possible for a student to be drilled through Level 2 assessment, no learner really understands mathematics at this level if they have not properly grasped proportion. But how might a course present proportion as a powerful idea in action?

What sort of problems?

Citizen Maths analyses ordinary contexts in which proportion is in fact powerful, such as when mixing, sharing, comparing and scaling. Another more complex situation is when trading off one quantity against another, which leads to the idea of inverse proportion. These five examples of how the powerful idea of proportion is brought into action then become the focus for designing meaningful problems, around, for example, mixing recipes or concrete, creating pie charts, looking for best buys, figuring out how the pinch gesture works in an iPhone, or deciding how many workers to deploy at the supermarket checkouts.

As well as working on problems with paper and pencil, tools such as calculators and spreadsheets are freely adopted. This is because at times it is more important to focus on the conceptual underpinning of the powerful ideas than on the details of calculation.

The course makes extensive use of applets on the Web or, when nothing suitable is available, applets that are specially designed for the course. The aim of the applets is to offer an on-screen manifestation of the powerful idea, which the learner can manipulate to gain a feel for how the powerful idea behaves. Such a holistic sense of the mathematical idea helps the learner to see it as a somewhat concrete object prior to working in more detail on computational aspects of the concept.

In a way, the approach helps to make the concept more visible, countering the trend for mathematics to become ever more hidden in the technological world. To this same end, we adopt Scratch, a programming environment, through which the learner ‘teaches’ the computer how to do the mathematics. In return, the mathematics not only becomes more visible but also the learner sees the pay-off for getting the mathematics (i.e. the program) correct, and is offered immediate system feedback when that is not the case.

Assessment

Citizen Maths may be help users gain fresh insights into mathematics before seeking a formal qualification; but it does not itself offer directly a qualification. Most learners, however, value feedback to help them to decide whether they are mastering the ideas. We therefore include simple assessment tasks within each activity.

The assessment tasks are offered as rough guidelines to help learners make a judgement about whether they are ready to move on. Sometimes, success in the assessment task may be fortunate and not reflect a good understanding of the ideas. Success in the assessment task and understanding of the tutor’s review of the activity may however confirm the learner’s general sense that they have understood the idea and that they are ready to move on.

Sometimes lack of success on the assessment task may result from a simple slip rather than a lack of understanding. Lack of success in the assessment task and not really being on top of the ideas when listening to the tutor’s review of the activity may confirm the learner’s general sense that they have not properly mastered the ideas and that they need to look at it again before moving on.

The key point here is that learners are in control of what they do on the course, rather than the course controlling them.

To sum things up

Citizen Maths is, like many other open online courses that are being offered around the world, an ambitious experiment. Structured according to the OECD’s PISA Assessment and Analytical Framework for Mathematics, Reading, Science, Problem Solving and Financial Literacy, Citizen Maths will provide a free and open means for self-motivated individuals to improve their mathematical capability at or around UK Level 2. It will give learners the feeling that they are in a one-to-one tutorial with a skilled teacher. We intend that from 2015 it will make use of adaptive learning, thereby enabling learners to take different paths through the course depending on progress. Rather than presenting mathematics as a sequence of apparently disconnected routines and procedures, it will engage learners in familiar activity to reveal the ‘maths inside’ and give access to powerful maths ideas in action.